#6545048

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#3267	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2091 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)(13,14,18,21,17,19,16,24)(15,20,23,28,22,26,25,27);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(21,24)(23,25);(6,9)(10,12)(13,14)(15,20)(16,21)(17,19)(18,24)(22,26)(23,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,15,17,22)(14,20,19,26)(16,25,18,23)(21,28,24,27);(7,11)(10,12)(13,16,17,18)(14,21,19,24)(15,23,22,25)(20,27,26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6452244);