#263544

order	512
nilpotency class	3
isoclinism	isoclinic class of #263527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62373	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

793 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 793 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170740);