#356620

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1635 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)(7,16)(8,13)(9,12)(10,15)(11,18)(14,23)(17,28)(19,26)(20,25)(21,27)(22,30)(24,32)(29,34)(31,36)(33,35);(5,7)(6,11,15,16)(8,20,22,17)(9,19)(10,18)(12,24,14,26)(13,28)(21,33,34,31)(23,32)(25,30)(27,36)(29,35);(1,2,4,3)(6,12)(9,23)(11,24)(13,27)(14,15)(16,26)(19,32)(21,34)(25,35)(28,36)(29,30)(31,33);(1,3,4,2)(5,8)(6,13)(7,17)(9,21)(10,22)(11,25)(12,27)(14,29)(15,30)(16,28)(18,20)(19,31)(23,34)(24,35)(26,36)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280827, #72280919, #72281232
maximal normal subgroup of	#72280827, #72280919, #72281232
maximal quotient of	#72313663

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263816);