#360725

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#385
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1347 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267921);