#359080

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1382 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916
maximal normal subgroup of	#72224916
maximal quotient of	#72224914

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266276);