#6371440

order	512
nilpotency class	3
isoclinism	isoclinic class of #60388
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

6073 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)(5,6,10,15,11,16,9,14)(7,17,20,12,21,26,19,27)(8,13,24,30,22,28,23,29)(18,31,34,25,32,36,33,35);(1,2,3,4)(5,7)(6,12)(8,18)(9,20)(10,19)(11,21)(13,25)(14,26)(15,17)(16,27)(22,32)(23,34)(24,33)(28,35)(29,36)(30,31);(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,25)(14,17)(15,26)(16,27)(22,32)(23,33)(24,34)(28,35)(29,31)(30,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,22)(12,25)(14,29)(15,30)(16,28)(17,31)(19,33)(20,34)(21,32)(26,36)(27,35);(8,22)(13,28)(18,32)(23,24)(25,35)(29,30)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,f^b=fi,h^b=hi,g^c=gi,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, 6278636);