#373823

order	512
nilpotency class	3
isoclinism	isoclinic class of #23912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1268 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)(5,6,10,14,11,15,9,22)(7,12,17,28,18,27,23,26)(8,19,21,13,20,31,24,30)(16,32,35,29,33,36,34,25);(1,2,4,3)(5,7)(6,12)(8,16)(9,23)(10,17)(11,18)(13,29)(14,28)(15,27)(19,32)(20,33)(21,35)(22,26)(24,34)(25,30)(31,36);(5,8,10,21,11,20,9,24)(6,13,14,31,15,30,22,19)(7,16,17,35,18,33,23,34)(12,25,28,32,27,29,26,36);(1,3,4,2)(8,20)(13,30)(16,33)(19,31)(21,24)(25,29)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=h,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,d^c=di,f^c=fi

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, 281019);