#6371982

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×#3500	C₂
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5046 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=gi,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, 6279178);