#432111

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1014 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 339307);