#361642

order	512
nilpotency class	3
isoclinism	isoclinic class of #220190
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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	C₂²×#577	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3546 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,g^c=gi,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, 268838);