#376235

order	512
nilpotency class	3
isoclinism	isoclinic class of #23900
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342922, #72342939
maximal normal subgroup of	#72342922, #72342939
maximal quotient of	#72343040

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283431);