#382641

order	512
nilpotency class	3
isoclinism	isoclinic class of #29592
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1310 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,c^b=cg,g^b=gi,h^b=hi,d^c=de

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289837);