#6539974

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

2131 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264113, #72264159, #72264300
maximal normal subgroup of	#72264113, #72264159, #72264300
maximal quotient of	#72264591

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6447170);