#6386196

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3511	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3158 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293392);