#233562

order	512
nilpotency class	3
isoclinism	isoclinic class of #233559
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1459 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1459 conjugacy classes of subgroups — too many to draw. Only its 190 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)(9,14)(10,13)(11,15)(12,16)(17,18,22,26)(19,23,29,37)(20,24,31,40)(21,32,34,25)(27,35,43,38)(28,41,33,36)(30,44,46,39)(42,48,45,47);(1,3)(2,6)(4,9)(5,10)(7,14)(8,13)(11,16)(12,15)(17,19,22,29)(18,23,26,37)(20,27,31,43)(21,33,34,28)(24,38,40,35)(25,41,32,36)(30,45,46,42)(39,47,44,48);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(17,20)(18,24)(19,27)(21,30)(22,31)(23,35)(25,39)(26,40)(28,42)(29,43)(32,44)(33,45)(34,46)(36,47)(37,38)(41,48);(4,11)(7,15)(9,16)(12,14)(17,21,22,34)(18,25,26,32)(19,28,29,33)(20,30,31,46)(23,36,37,41)(24,39,40,44)(27,42,43,45)(35,47,38,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=dh,f^b=fi,d^c=di,e^c=eh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285456, #72285462, #72285528
maximal normal subgroup of	#72285456, #72285462, #72285528

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140758);