#364543

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
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×#3018	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214895, #72311922, #72311961, …
maximal normal subgroup of	#72214895, #72311922, #72311961, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271739);