#423864

order	512
nilpotency class	3
isoclinism	isoclinic class of #422715
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1279 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=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, 331060);