#380431

order	512
nilpotency class	3
isoclinism	isoclinic class of #30056
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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)(5,6,11,15,12,16,10,9)(7,17,21,13,22,19,20,31)(8,23,27,14,28,29,26,25)(18,33,34,36,35,30,24,32);(5,7)(6,13)(8,24)(9,19)(10,21)(11,20)(12,22)(14,32)(15,17)(16,31)(18,26)(23,33)(25,36)(27,35)(28,34)(29,30);(5,8,12,28)(6,14,16,25)(7,18,22,35)(9,29,15,23)(10,27,11,26)(13,30,31,33)(17,32,19,36)(20,34,21,24);(5,9,10,16,12,15,11,6)(7,19,21,31,22,17,20,13)(8,25,26,29,28,14,27,23)(18,30,34,32,35,33,24,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 287627);