#380849

order	512
nilpotency class	3
isoclinism	isoclinic class of #29730
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

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

1042 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 288045);