#6591907

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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2045 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,d^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, 6499103);