#354851

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1486 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=h,h^2=i,b^a=be,c^a=cf,f^a=fi,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240734, #72240806, #72264458
maximal normal subgroup of	#72240734, #72240806, #72264458

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262047);