#6596897

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×#3282	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2251 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)(7,10)(8,11)(9,12)(13,14,19,24)(15,20,25,21)(16,22,26,28)(17,23,18,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,18)(17,26)(19,25)(21,24)(22,27)(23,28);(7,12)(9,10)(13,16,19,26)(14,21,24,20)(15,17,25,18)(22,23,28,27);(5,7)(6,10)(8,12)(9,11)(13,16)(14,22)(15,17)(18,25)(19,26)(20,23)(21,27)(24,28);(7,12)(9,10)(13,17,19,18)(14,23,24,27)(15,16,25,26)(20,22,21,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=i,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, 6504093);