#378048

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
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

1075 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,d^2=fh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,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, 285244);