#429182

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1150 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)(7,10)(8,12)(9,11)(13,14,18,23)(15,25,29,38)(16,30,32,21)(17,24,19,22)(20,27,35,28)(26,40,43,39)(31,36,33,37)(34,42,44,41);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,29)(14,20,23,35)(16,26,32,43)(17,27,19,28)(21,34,30,44)(22,38,24,25)(31,41,33,42)(36,40,37,39);(2,4)(6,11)(10,12)(13,16,18,32)(14,21,23,30)(15,26,29,43)(17,31,19,33)(20,34,35,44)(22,36,24,37)(25,39,38,40)(27,41,28,42);(5,8,9,7)(6,12,11,10)(14,22)(15,27)(17,19)(20,35)(21,36)(23,24)(26,41)(28,29)(30,37)(31,33)(34,44)(42,43)

Polycyclic presentation:

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

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, 336378);