#422513

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1186 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 329709);