#393997

order	512
nilpotency class	3
isoclinism	isoclinic class of #393982
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

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

3947 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:

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

Polycyclic presentation:

2^9|a^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,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, 301193);