#6371981

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342612
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3499	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4944 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fhi,c^2=g,h^2=i,b^a=bf,c^a=cg,f^a=fi,f^b=fi,h^b=hi,e^d=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279177);