#374382

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1723 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)(7,15)(8,12,20,28)(9,13)(10,14)(11,17)(16,33,34,25)(18,31)(19,36,35,27)(21,29,24,32)(22,30)(23,26);(1,2,4,3)(5,7,9,17)(6,11,13,15)(8,19,20,35)(10,23,22,18)(12,27,28,36)(14,31,30,26)(16,24,34,21)(25,32,33,29);(5,8)(6,12,13,28)(7,16)(9,20)(10,24)(11,25,15,33)(14,32,30,29)(17,34)(18,35)(19,23)(21,22)(26,36,31,27);(1,3,4,2)(6,13)(11,15)(12,28)(14,30)(25,33)(26,31)(27,36)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi,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, 281578);