#394678

order	512
nilpotency class	3
isoclinism	isoclinic class of #394382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2185 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)(5,6,11,16)(7,17,20,32)(8,21,24,13)(9,15,10,14)(12,26,28,19)(18,34,22,33)(23,30,25,31)(27,35,29,36);(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,29)(14,32)(15,17)(16,28)(18,24)(21,27)(23,35)(25,36)(30,33)(31,34);(2,4)(5,8,9,23,11,24,10,25)(6,13,14,30,16,21,15,31)(7,18,19,35,20,22,26,36)(12,27,17,33,28,29,32,34);(5,7)(6,12)(8,18)(9,19)(10,26)(11,20)(13,27)(14,17)(15,32)(16,28)(21,29)(22,24)(23,35)(25,36)(30,33)(31,34)

Polycyclic presentation:

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

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, 301874);