#417194

order	512
nilpotency class	3
isoclinism	isoclinic class of #394257
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324390);