#251919

order	512
nilpotency class	3
isoclinism	isoclinic class of #251883
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

956 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 956 conjugacy classes of subgroups — too many to draw. Only its 126 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,18,22,26)(19,23,29,36)(20,24,32,38)(21,25,33,39)(27,34,30,42)(28,35,41,47)(31,44,45,37)(40,48,43,46);(1,3,5,10)(2,6,8,12)(4,9,11,15)(7,13,14,16)(17,19)(18,23)(20,30,32,27)(21,28)(22,29)(24,34,38,42)(25,35)(26,36)(31,43,45,40)(33,41)(37,46,44,48)(39,47);(4,11)(6,12)(7,14)(9,15)(17,20)(18,24)(19,27)(21,31)(22,32)(23,34)(25,37)(26,38)(28,40)(29,30)(33,45)(35,46)(36,42)(39,44)(41,43)(47,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,21)(18,25)(19,28)(20,31)(22,33)(23,35)(24,37)(26,39)(27,40)(29,41)(30,43)(32,45)(34,46)(36,47)(38,44)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285063, #72285236, #72285348, …
maximal normal subgroup of	#72285063, #72285236, #72285348, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159115);