#395238

order	512
nilpotency class	3
isoclinism	isoclinic class of #395238
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	#23409	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1706 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)(7,13)(8,21)(9,24)(10,17)(11,16)(12,19)(14,26)(15,22)(18,32)(20,28)(23,29)(25,30)(27,34)(31,35)(33,36);(5,7,10,19,11,8,22,20)(6,12,15,13,16,28,17,21)(9,18,25,33,26,23,35,34)(14,27,30,29,24,36,31,32);(2,4)(5,8)(6,13)(7,11)(9,23)(10,20)(12,17)(14,29)(15,28)(16,21)(18,26)(19,22)(24,32)(25,34)(27,31)(30,36)(33,35);(5,9,10,25,11,26,22,35)(6,14,15,30,16,24,17,31)(7,18,19,33,8,23,20,34)(12,27,13,29,28,36,21,32)

Polycyclic presentation:

2^9|b^2=e,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,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, 302434);