#6597127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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

2290 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:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264210, #72264241, #72264263
maximal normal subgroup of	#72264210, #72264241, #72264263

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504323);