#777310

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

2381 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)(7,12)(8,9)(10,11)(13,14,18,20)(15,16,17,19)(22,25)(26,28);(6,11)(9,12)(13,15,18,17)(14,19,20,16)(22,25)(26,28);(5,7,10,8)(6,12,11,9)(13,16)(14,15)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,10,7)(6,12,11,9)(13,17,18,15)(14,16,20,19);(5,9,10,12)(6,8,11,7)(13,16)(14,17)(15,20)(18,19)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=fgh,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, 684506);