#6380839

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=dfi,d^c=di,e^c=ei

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, 6288035);