#401069

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1754 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,c^2=i,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,h^c=hi,e^d=ei,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, 308265);