#411128

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
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 ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 318324);