#6459653

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3268 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:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gh,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,d^c=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366849);