#364525

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂×#3025	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2430 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314334, #72314335, #72314348
maximal normal subgroup of	#72314334, #72314335, #72314348
maximal quotient of	#72314868

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271721);