#394326

order	512
nilpotency class	3
isoclinism	isoclinic class of #394044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2604 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,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^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	#72314237, #72314267, #72314334
maximal normal subgroup of	#72314237, #72314267, #72314334
maximal quotient of	#72314513

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301522);