#6598349

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3319	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1924 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505545);