#6445127

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2225 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)(7,10)(8,9)(11,12)(15,21)(16,24)(18,25)(20,26)(22,27)(23,28);(1,2)(3,4)(6,9)(10,12)(13,14)(15,22)(16,20)(17,19)(18,23)(21,27)(24,26)(25,28);(2,4)(13,15,16,21,17,23,24,28)(14,18,20,25,19,22,26,27);(2,4)(15,23)(18,22)(21,28)(25,27);(5,7,8,11)(6,10,9,12)(14,19)(18,22)(20,26)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307088, #72307249, #72307338
maximal normal subgroup of	#72307088, #72307249, #72307338

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6352323);