#383124

order	512
nilpotency class	3
isoclinism	isoclinic class of #222272
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2077 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290320);