#6372212

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,e^d=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279408);