#394324

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3027	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2620 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)(6,12)(9,10)(13,17)(14,25)(15,16)(18,32)(19,20)(22,24)(26,35)(27,28)(29,30)(31,36);(1,2,3,4)(5,6,9,12,11,16,10,15)(7,17,19,13,21,27,20,28)(8,14,22,25,23,29,24,30)(18,31,33,26,32,35,34,36);(5,7)(6,13)(8,18)(9,20)(10,19)(11,21)(12,17)(14,26)(15,27)(16,28)(22,34)(23,32)(24,33)(25,31)(29,36)(30,35);(5,8)(6,14)(7,18)(9,22)(10,24)(11,23)(12,25)(13,26)(15,30)(16,29)(17,31)(19,33)(20,34)(21,32)(27,35)(28,36)

Polycyclic presentation:

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

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, 301520);