#698356

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 605552);