#844828

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fgi,d^2=h,e^2=hi,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, 752024);