#356110

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₂²⋊C₄
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2263 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)(3,4)(5,6,9,12,11,16,10,15)(7,17,20,13,21,27,19,28)(8,14,22,25,24,30,23,29)(18,31,33,26,34,35,32,36);(6,12,16,15)(8,22,24,23)(13,17,28,27)(18,32,34,33);(2,4)(5,7)(6,13)(8,18)(9,19)(10,20)(11,21)(12,17)(14,26)(15,27)(16,28)(22,32)(23,33)(24,34)(25,31)(29,35)(30,36);(5,8)(6,14)(7,18)(9,22)(10,23)(11,24)(12,25)(13,26)(15,29)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224693, #72312189, #72312195
maximal normal subgroup of	#72224693, #72312189, #72312195

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263306);