#381854

order	512
nilpotency class	3
isoclinism	isoclinic class of #29602
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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1302 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)(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,26)(21,27)(23,25)(24,30)(28,31)(29,32);(10,13)(14,16)(17,19,22,18)(20,28,30,32)(21,27,23,25)(24,31,26,29);(2,4)(11,15)(14,16)(17,20,22,30)(18,24,19,26)(21,31,23,29)(25,28,27,32);(5,7,8,6)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,25,19,27)(20,29,30,31)(24,32,26,28)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=efhi,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, 289050);