#6558117

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2594 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)(13,14,18,21)(15,22,20,17)(16,19,25,27)(23,28,24,26);(5,6)(7,10)(8,9)(11,12)(13,15,17,14,18,20,22,21)(16,24,26,27,25,23,28,19);(1,2,3,4)(6,9)(10,12)(13,16,18,25)(14,19,21,27)(15,23,20,24)(17,26,22,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,23)(17,26)(18,25)(20,24)(21,27)(22,28);(7,11)(10,12)(16,25)(19,27)(23,24)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6465313);