#394092

order	512
nilpotency class	3
isoclinism	isoclinic class of #394090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3948 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)(5,6,10,16)(7,17,21,13)(8,14,24,30)(9,12,11,15)(18,28,20,25)(19,27,34,31)(22,26,23,29)(32,35,33,36);(2,4)(6,12)(7,18)(9,11)(14,26)(15,16)(19,32)(20,21)(22,23)(25,28)(29,30)(33,34)(35,36);(5,7)(6,13)(8,19)(9,18)(10,21)(11,20)(12,25)(14,27)(15,28)(16,17)(22,32)(23,33)(24,34)(26,35)(29,36)(30,31);(5,8)(6,14)(7,19)(9,22)(10,24)(11,23)(12,26)(13,27)(15,29)(16,30)(17,31)(18,32)(20,33)(21,34)(25,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279287, #72314237, #72314334
maximal normal subgroup of	#72279287, #72314237, #72314334
maximal quotient of	#72279295, #72279713, #72280491, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301288);