#417965

order	512
nilpotency class	3
isoclinism	isoclinic class of #417544
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1985 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)(7,16,19,20)(12,17,26,30)(18,31,34,35)(25,32,36,33);(5,6)(7,17,19,30)(8,13)(9,23)(10,14)(11,15)(12,20,26,16)(18,33,34,32)(21,28)(22,29)(24,27)(25,31,36,35);(5,7)(6,12)(8,18)(9,20)(10,16)(11,19)(13,25)(14,30)(15,26)(17,23)(21,35)(22,34)(24,31)(27,33)(28,32)(29,36);(2,4)(5,8,11,22)(6,13,15,29)(7,18,19,34)(9,21,10,24)(12,25,26,36)(14,27,23,28)(16,31,20,35)(17,32,30,33)

Polycyclic presentation:

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

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, 325161);