#6667909

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2941 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:

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

Polycyclic presentation:

2^9|b^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282013, #72282497, #72282746
maximal normal subgroup of	#72282013, #72282497, #72282746

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6575105);