#6684774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2377 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)(5,6,8,9)(7,11,12,10)(13,14,17,21)(15,22,24,19)(16,20,27,18)(23,28,25,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,24)(18,23)(20,25)(21,22)(27,28);(1,2,3,4)(6,9)(10,11)(13,16,17,27)(14,20,21,18)(15,22,24,19)(23,26,25,28);(5,7,8,12)(6,10,9,11)(14,21)(16,27)(19,22)(26,28);(14,21)(15,23)(18,20)(19,28)(22,26)(24,25)

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,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, #72256878, #72256954, …
maximal normal subgroup of	#72256847, #72256878, #72256954, …
maximal quotient of	#72264374, #72264717

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6591970);