#365365

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	C₂×#385
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1510 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)(5,6,9,14,10,13,21,27)(7,11,17,24,16,23,31,34)(8,12,20,25,19,26,29,35)(15,28,30,36,18,22,32,33);(2,4)(5,7,9,17,10,16,21,31)(6,11,14,24,13,23,27,34)(8,18,20,32,19,15,29,30)(12,22,25,33,26,28,35,36);(5,8)(6,12)(7,15,16,18)(9,20)(10,19)(11,22,23,28)(13,26)(14,25)(17,30,31,32)(21,29)(24,33,34,36)(27,35);(2,4)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,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	#72322778, #72322900, #72342923, …
maximal normal subgroup of	#72322778, #72322900, #72342923, …
maximal quotient of	#72343078

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272561);