#394084

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	#62852	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4046 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:

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

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^c=hi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240222, #72314237, #72314334
maximal normal subgroup of	#72240222, #72314237, #72314334
maximal quotient of	#72240223, #72260824, #72260865, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301280);