#414152

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
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:

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

Polycyclic presentation:

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

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, 321348);