#777302

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2519 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)(5,6,10,9)(7,8,12,11)(14,18)(19,20)(21,22)(23,24)(25,26)(27,28);(5,7,10,12)(6,11,9,8)(13,14)(15,19)(16,20)(17,18)(22,26)(23,28);(5,8)(6,12)(7,9)(10,11)(13,15,17,16)(14,19,18,20)(21,23,25,28)(22,27,26,24);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,24)(22,23)(25,27)(26,28);(5,9,10,6)(7,11,12,8)(13,16,17,15)(14,19,18,20)(21,22)(23,27)(24,28)(25,26)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=fgh,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, 684498);