#376687

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1337 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320982, #72321148, #72342922, …
maximal normal subgroup of	#72320982, #72321148, #72342922, …
maximal quotient of	#72343022

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283883);