#6597014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2246 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)(13,14,18,22)(15,19,17,20)(16,24,26,21)(23,28,25,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,17)(14,19,22,20)(16,25,26,23)(21,28,24,27);(7,12)(9,10)(14,20)(15,17)(16,26)(19,22)(21,27)(24,28);(5,7)(6,10)(8,12)(9,11)(13,16,18,26)(14,21,22,24)(15,23,17,25)(19,27,20,28);(7,12)(9,10)(16,26)(21,24)(23,25)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,d^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=ehi,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504210);