#778466

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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2158 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)(7,8)(9,10)(11,12)(13,14,17,16)(15,18,20,19)(21,22,26,28)(23,24,25,27);(6,10)(8,11)(13,14)(15,19)(16,17)(18,20)(21,23,26,25)(22,24,28,27);(13,15)(14,18)(16,19)(17,20)(21,24,26,27)(22,23,28,25);(1,2)(3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,19)(16,18)(17,20)(21,22)(23,24)(25,27)(26,28);(5,8,9,11)(6,7,10,12)(13,16,17,14)(15,19,20,18)(21,25,26,23)(22,27,28,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=fh,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, 685662);