#380102

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287298);