#6459664

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437770
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 ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3387 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)(7,9)(8,10)(11,12)(13,14,19,16)(15,22,23,28)(17,21,18,27)(20,24,26,25);(5,7)(6,9)(8,11)(10,12)(13,15,18,25,19,23,17,24)(14,20,21,22,16,26,27,28);(1,2,3,4)(6,10)(7,11)(13,16,19,14)(15,20,23,26)(17,21,18,27)(22,24,28,25);(7,11)(9,12)(15,23)(20,26)(22,28)(24,25);(13,17,19,18)(14,21,16,27)(15,24,23,25)(20,22,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280757, #72281055, #72310690, …
maximal normal subgroup of	#72280757, #72281055, #72310690, …
maximal quotient of	#72313058, #72313086

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366860);