#398439

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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1556 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)(7,18)(8,14)(9,23)(10,27)(11,16)(12,17)(13,20)(15,26)(19,30)(21,32)(22,29)(24,31)(25,36)(28,34)(33,35);(5,7,10,20,12,8,11,22)(6,13,16,18,17,29,27,14)(9,19,24,33,26,21,25,34)(15,28,31,32,23,35,36,30);(1,2,3,4)(5,8,12,7)(6,14,17,18)(9,21,26,19)(10,22,11,20)(13,27,29,16)(15,30,23,32)(24,34,25,33)(28,36,35,31);(5,9,10,24,12,26,11,25)(6,15,16,31,17,23,27,36)(7,19,20,33,8,21,22,34)(13,28,18,32,29,35,14,30)

Polycyclic presentation:

2^9|b^2=e,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, 305635);