#6598409

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×#3305	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6505605);