#407700

order	512
nilpotency class	3
isoclinism	isoclinic class of #393940
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1152 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 314896);