#493066

order	512
nilpotency class	4
isoclinism	isoclinic class of #33388
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1166 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)(7,18)(8,13)(9,15)(10,27)(11,20)(12,17)(14,26)(16,25)(19,32)(21,28)(22,30)(23,36)(24,33)(29,34)(31,35);(5,7,10,20,11,14,9,6,12,15,26,16,25,27,18,17)(8,19,23,33,24,29,22,13,21,30,34,31,35,36,32,28);(1,3,4,2)(5,8)(6,13)(7,19)(9,22)(10,23)(11,24)(12,21)(14,29)(15,30)(16,31)(17,28)(18,32)(20,33)(25,35)(26,34)(27,36);(8,21)(13,28)(19,30)(22,32)(23,34)(24,35)(29,36)(31,33)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=hi,g^2=h,h^2=i,b^a=be,e^a=eh,g^a=gh,h^a=hi,e^b=ei,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239482, #72239524, #72239548, …
maximal normal subgroup of	#72239482, #72239524, #72239548, …
maximal quotient of	#72319291, #72320010, C₂×#493066

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400262);