#362303

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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

822 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:

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

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,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269499);