#471595

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

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

1242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,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, 378791);