#362302

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

836 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eg,c^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226664, #72226668, #72226670
maximal normal subgroup of	#72226664, #72226668, #72226670

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269498);