#358057

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1774 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 265253);