#611787

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2708 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

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, 518983);