#6685197

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2179 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,18)(15,20,25,24)(16,21,17,26)(22,27,28,23);(6,9)(7,11)(13,15)(14,20)(16,23)(17,27)(18,24)(19,25)(21,22)(26,28);(1,2,3,4)(13,16)(14,21)(15,22)(17,19)(18,26)(20,27)(23,24)(25,28);(6,9)(10,12)(13,17,19,16)(14,21,18,26)(15,23,25,27)(20,28,24,22);(5,7,8,11)(6,10,9,12)(14,18)(15,24)(20,25)(21,26)(22,23)(27,28)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256847, #72256854, #72257027, …
maximal normal subgroup of	#72256847, #72256854, #72257027, …
maximal quotient of	#72264453, #72264491, #72264695

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592393);