#373675

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2240 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)(3,4)(9,10,15,20,16,21,14,30)(11,22,13,17,24,19,29,32)(12,18,27,34,28,35,26,39)(23,36,25,31,37,33,38,40);(5,6)(7,8)(9,11)(10,17)(12,23)(13,15)(14,29)(16,24)(18,31)(19,20)(21,32)(22,30)(25,27)(26,38)(28,37)(33,34)(35,40)(36,39);(2,4)(6,8)(9,12,15,27,16,28,14,26)(10,18,20,34,21,35,30,39)(11,23,13,25,24,37,29,38)(17,31,19,33,32,40,22,36);(9,13)(10,19)(11,14)(12,25)(15,24)(16,29)(17,30)(18,33)(20,32)(21,22)(23,26)(27,37)(28,38)(31,39)(34,40)(35,36)

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280871);