#483110

order	512
nilpotency class	4
isoclinism	isoclinic class of #482075
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1079 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,3,6,4,5,7,8)(9,10,14,20,15,18,30,36,16,21,28,38,12,27,13,19)(11,22,25,35,23,37,24,32,26,39,29,17,31,33,40,34);(2,5)(3,7)(9,11)(10,17)(12,23)(13,29)(14,24)(15,31)(16,26)(18,32)(19,37)(20,22)(21,35)(25,30)(27,34)(28,40)(33,36)(38,39);(2,5)(6,8);(9,12,16,15)(10,18,21,27)(11,23,26,31)(13,28,30,14)(17,32,35,34)(19,20,36,38)(22,33,39,37)(24,29,40,25)

Polycyclic presentation:

2^9|a^2=f,d^2=i,e^2=hi,f^2=gh,h^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,e^b=eh,f^b=fgh,h^b=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72309624, #72309670, #72309682, …
maximal normal subgroup of	#72309624, #72309670, #72309682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390306);