#493358

order	512
nilpotency class	4
isoclinism	isoclinic class of #480250
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	C₂×#3449	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

3452 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)(7,15)(8,12)(9,21,22,13)(10,14)(11,18,24,31)(16,27)(17,28)(19,32,33,25)(20,26)(23,30,34,36)(29,35);(5,7)(6,11)(8,16)(9,18)(10,17)(12,23)(13,15)(14,24)(19,30)(20,29)(21,28)(22,31)(25,27)(26,34)(32,35)(33,36);(2,4)(5,8,10,20)(6,12,14,26)(7,16,17,29)(9,19,22,33)(11,23,24,34)(13,25,21,32)(15,27,28,35)(18,30,31,36);(7,17)(11,24)(15,28)(16,29)(18,31)(23,34)(27,35)(30,36)

Polycyclic presentation:

2^9|a^2=g,c^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280213, #72317594, #72317615
maximal normal subgroup of	#72280213, #72317594, #72317615
maximal quotient of	#72317719, #72317927

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400554);