#425754

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1373 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,g^d=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 332950);