#377008

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1085 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)(3,8)(4,6)(5,7)(9,10,14,20,15,19,29,35)(11,16,24,33,23,32,37,40)(12,17,27,34,26,13,18,28)(21,22,31,36,25,30,38,39);(1,3,5,4)(2,6,7,8)(9,11)(10,16)(12,25)(13,22)(14,24)(15,23)(17,30)(18,31)(19,32)(20,33)(21,26)(27,38)(28,36)(29,37)(34,39)(35,40);(1,4,5,3)(2,6,7,8)(9,12)(10,17)(11,21,23,25)(13,19)(14,27)(15,26)(16,30,32,22)(18,29)(20,34)(24,31,37,38)(28,35)(33,39,40,36);(2,7)(6,8)(9,13,14,28,15,17,29,34)(10,18,20,12,19,27,35,26)(11,22,24,36,23,30,37,39)(16,31,33,25,32,38,40,21)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284204);