#365664

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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)(3,7)(4,8)(5,6)(9,10,14,18,15,19,13,27)(11,16,22,32,21,30,28,31)(12,17,26,35,24,34,25,39)(20,36,23,33,37,29,38,40);(2,6)(7,8)(9,11,14,22,15,21,13,28)(10,16,18,32,19,30,27,31)(12,23,26,37,24,38,25,20)(17,33,35,29,34,40,39,36);(1,3,5,4)(2,7,6,8)(9,12)(10,17)(11,20,21,37)(13,25)(14,26)(15,24)(16,29,30,36)(18,35)(19,34)(22,23,28,38)(27,39)(31,33,32,40);(1,4,5,3)(2,7,6,8)(12,24)(17,34)(20,37)(23,38)(25,26)(29,36)(33,40)(35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272860);