#790472

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2654 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 697668);