#398441

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1588 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 305637);