#376908

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	C₄×#120
prime radical	#23399	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1482 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)(5,6,11,16,12,17,10,31)(7,13,20,34,21,24,32,33)(8,22,27,35,26,19,25,14)(9,15,29,23,30,36,28,18);(2,4)(5,7)(6,13)(8,23)(9,19)(10,32)(11,20)(12,21)(14,29)(15,25)(16,34)(17,24)(18,26)(22,30)(27,36)(28,35)(31,33);(2,4)(5,8,10,25,12,26,11,27)(6,14,16,22,17,35,31,19)(7,18,20,15,21,23,32,36)(9,24,28,34,30,13,29,33);(2,4)(5,9,11,29,12,30,10,28)(6,15,16,23,17,36,31,18)(7,19,20,14,21,22,32,35)(8,24,27,33,26,13,25,34)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=h,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, 284104);