#777308

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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2385 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fgh,c^2=gh,d^2=f,e^2=fg,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, 684504);