#6386173

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×#3494	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3188 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6293369);