#372574

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1412 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)(7,8)(9,10)(11,15)(12,13)(14,16)(17,18,21,25,22,19,23,27)(20,24,29,31,28,26,30,32);(2,4)(5,6)(7,8)(10,13)(14,15)(17,19,21,27,22,18,23,25)(20,26,29,32,28,24,30,31);(2,4)(6,8)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32);(2,4)(9,11,12,16)(10,14,13,15)(18,19)(20,28)(25,27)(29,30)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=efg,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279770);