#6370661

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1956 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=fg,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277857);