#368025

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1680 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250174, #72250604, #72256881, …
maximal normal subgroup of	#72250174, #72250604, #72256881, …
maximal quotient of	C₂×#368025, #72343028

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275221);