#395448

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62853	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3504 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:

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

Polycyclic presentation:

2^9|a^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,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, 302644);