#406181

order	512
nilpotency class	3
isoclinism	isoclinic class of #394189
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1682 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=fi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313377);