#392805

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1529 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342922, #72342942, #72342948
maximal normal subgroup of	#72342922, #72342942, #72342948
maximal quotient of	#72343044, #72343141

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300001);