#825696

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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

3486 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=i,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, 732892);