#404016

order	512
nilpotency class	3
isoclinism	isoclinic class of #396094
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	#23444	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2735 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)(7,11)(8,12)(9,20)(10,14)(13,21)(15,22)(16,30)(17,24)(18,32)(19,27)(23,31)(25,29)(26,33)(28,35)(34,36);(5,7,10,17)(6,11,14,24)(8,15,19,29)(9,16,21,31)(12,25,27,22)(13,23,20,30)(18,28,33,36)(26,35,32,34);(2,4)(5,8)(6,12)(7,15)(9,18)(10,19)(11,22)(13,26)(14,27)(16,28)(17,29)(20,32)(21,33)(23,34)(24,25)(30,35)(31,36);(5,9)(6,13,14,20)(7,16)(8,18)(10,21)(11,23,24,30)(12,26,27,32)(15,28)(17,31)(19,33)(22,34,25,35)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237666, #72262658, #72262670, …
maximal normal subgroup of	#72237666, #72262658, #72262670, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311212);