#6383857

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61038	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2554 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,e^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ci,d^b=df,f^b=fi,f^d=fi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#2955, #72286208, #72286245
maximal normal subgroup of	D₄×#2955, #72286208, #72286245
maximal quotient of	#72286245, #72287242, #72287402, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6291053);