#509807

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

817 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343483, #72343812, #72343853
maximal normal subgroup of	#72343483, #72343812, #72343853

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417003);