#6372031

order	512
nilpotency class	3
isoclinism	isoclinic class of #60470
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#576	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5028 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)(5,6,11,16,12,15,10,7)(8,19,22,28,23,17,21,13)(9,14,26,30,24,29,25,18)(20,32,35,36,33,31,34,27);(1,2,3,4)(5,7,11,15,12,16,10,6)(8,17,22,28,23,19,21,13)(9,18,26,29,24,30,25,14)(20,31,35,36,33,32,34,27);(5,8,11,22,12,23,10,21)(6,13,7,17,15,28,16,19)(9,20,26,35,24,33,25,34)(14,27,18,31,29,36,30,32);(5,9)(6,14)(7,18)(8,20)(10,25)(11,26)(12,24)(13,27)(15,29)(16,30)(17,31)(19,32)(21,34)(22,35)(23,33)(28,36);(9,24)(14,29)(18,30)(20,33)(25,26)(27,36)(31,32)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=gi,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,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, 6279227);