#455950

order	512
nilpotency class	3
isoclinism	isoclinic class of #455564
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23393	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1176 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)(7,11)(8,13)(9,14)(10,22)(12,21)(15,16)(17,26)(18,32)(19,28)(20,31)(23,27)(24,30)(25,33)(29,35)(34,36);(5,7,16,18,9,19,10,23)(6,11,22,27,14,28,15,32)(8,17,24,33,20,34,21,35)(12,29,31,36,30,25,13,26);(5,8,9,20)(6,12,14,30)(7,17)(10,24,16,21)(11,25)(13,15,31,22)(18,35)(19,34)(23,33)(26,32)(27,36)(28,29);(2,4)(5,8)(6,13)(7,17)(9,20)(10,21)(11,26)(12,22)(14,31)(15,30)(16,24)(18,33)(19,34)(23,35)(25,32)(27,29)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363146);