#509304

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

679 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343325, #72343836, #72343845
maximal normal subgroup of	#72343325, #72343836, #72343845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416500);