#6370706

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	C₂²×C₄	C₂⁵
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

17126 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)(5,6,10,16,11,15,8,13)(7,17,18,30,20,12,24,25)(9,14,23,29,22,28,21,27)(19,31,32,36,33,26,34,35);(5,7)(6,12)(8,18)(9,19)(10,24)(11,20)(13,25)(14,26)(15,17)(16,30)(21,32)(22,33)(23,34)(27,35)(28,31)(29,36);(2,4)(5,8,11,10)(6,13,15,16)(7,18,20,24)(9,21,22,23)(12,25,17,30)(14,27,28,29)(19,32,33,34)(26,35,31,36);(5,9)(6,14)(7,19)(8,21)(10,23)(11,22)(12,26)(13,27)(15,28)(16,29)(17,31)(18,32)(20,33)(24,34)(25,35)(30,36);(9,22)(14,28)(19,33)(21,23)(26,31)(27,29)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,c^2=i,h^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	#72306293, #72306335, #72306401, …
maximal normal subgroup of	#72306293, #72306335, #72306401, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277902);