#356967

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	C₂³×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1490 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)(5,6,9,15,11,16,10,26)(7,17,21,34,20,12,22,29)(8,14,23,31,24,30,25,13)(18,28,35,27,19,32,36,33);(5,7,10,22,11,20,9,21)(6,12,15,29,16,17,26,34)(8,18,23,35,24,19,25,36)(13,27,30,28,31,33,14,32);(1,2,4,3)(5,8)(6,13)(7,18)(9,25)(10,23)(11,24)(12,27)(14,26)(15,30)(16,31)(17,33)(19,20)(21,36)(22,35)(28,29)(32,34);(1,3,4,2)(5,8)(6,14)(7,19)(9,23)(10,25)(11,24)(12,28)(13,26)(15,31)(16,30)(17,32)(18,20)(21,36)(22,35)(27,29)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312215, #72312252, #72312295
maximal normal subgroup of	#72312215, #72312252, #72312295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264163);