#471976

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1672 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

(6,7)(10,13)(14,16)(17,18,23,29,22,20,21,25)(19,24,30,32,27,26,28,31);(1,2,3,4)(5,6,8,7)(9,10,12,13)(11,14,15,16)(17,19,22,27)(18,24,20,26)(21,28,23,30)(25,31,29,32);(2,4)(6,7)(11,15)(14,16)(17,20,22,18)(19,26,27,24)(21,25,23,29)(28,31,30,32);(5,7,8,6)(9,11,12,15)(10,14,13,16)(19,27)(24,26)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=eghi,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,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, 379172);