#6566258

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3345	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ei,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308713, #72308845
maximal normal subgroup of	#72308713, #72308845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6473454);