#376142

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#2017	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1520 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72214895, #72264612, #72312221
maximal normal subgroup of	#72214895, #72264612, #72312221

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283338);