#395979

order	512
nilpotency class	3
isoclinism	isoclinic class of #394127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,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, 303175);