#494474

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1452 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)(5,6,10,15,26,20,9,25,12,16,27,19,11,7,17,14)(8,13,23,30,35,32,22,36,21,28,34,33,24,18,31,29);(5,7,11,6,12,20,26,16)(8,18,24,13,21,32,35,28)(9,25,10,19,17,14,27,15)(22,36,23,33,31,29,34,30);(2,4)(5,8)(6,13)(7,18)(9,22)(10,23)(11,24)(12,21)(14,29)(15,30)(16,28)(17,31)(19,33)(20,32)(25,36)(26,35)(27,34);(8,21)(13,28)(18,32)(22,31)(23,34)(24,35)(29,36)(30,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225127, #72249872, #72249897, …
maximal normal subgroup of	#72225127, #72249872, #72249897, …
maximal quotient of	#72225030, #72286291, #72287176, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401670);