#372737

order	512
nilpotency class	3
isoclinism	isoclinic class of #222240
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3041	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,d^2=efg,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279933);