#6370696

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342574
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

17218 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=i,e^2=i,b^a=bf,c^a=cg,f^a=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306335, #72306404, #72306468, …
maximal normal subgroup of	#72306335, #72306404, #72306468, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277892);