#6370694

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

17274 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)(7,15,18,11)(8,12)(9,13)(10,14)(16,27,30,22)(17,28,31,23)(19,24)(20,25)(21,26)(29,35,36,33)(32,34);(5,7)(6,11)(8,16)(9,17)(10,18)(12,22)(13,23)(14,15)(19,29)(20,30)(21,31)(24,33)(25,27)(26,28)(32,36)(34,35);(2,4)(5,8,10,20)(6,12,14,25)(7,16,18,30)(9,19,21,32)(11,22,15,27)(13,24,26,34)(17,29,31,36)(23,33,28,35);(5,9)(6,13)(7,17)(8,19)(10,21)(11,23)(12,24)(14,26)(15,28)(16,29)(18,31)(20,32)(22,33)(25,34)(27,35)(30,36);(9,21)(13,26)(17,31)(19,32)(23,28)(24,34)(29,36)(33,35)

Polycyclic presentation:

2^9|a^2=h,c^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	#72306291, #72306401, #72306404, …
maximal normal subgroup of	#72306291, #72306401, #72306404, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277890);