#376867

order	512
nilpotency class	3
isoclinism	isoclinic class of #24404
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	#23443	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1543 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)(5,6,10,16,11,15,25,31)(7,12,20,29,19,28,33,36)(8,13,23,30,22,9,14,24)(17,18,27,32,21,26,34,35);(2,4)(5,7)(6,12)(8,21)(9,18)(10,20)(11,19)(13,26)(14,27)(15,28)(16,29)(17,22)(23,34)(24,32)(25,33)(30,35)(31,36);(2,4)(5,8)(6,13)(7,17,19,21)(9,15)(10,23)(11,22)(12,26,28,18)(14,25)(16,30)(20,27,33,34)(24,31)(29,35,36,32);(2,4)(5,9,10,24,11,13,25,30)(6,14,16,8,15,23,31,22)(7,18,20,32,19,26,33,35)(12,27,29,21,28,34,36,17)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,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	#72321557, #72322108, #72342923, …
maximal normal subgroup of	#72321557, #72322108, #72342923, …
maximal quotient of	#72343053

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284063);