#422833

order	512
nilpotency class	3
isoclinism	isoclinic class of #420420
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60644	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1910 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)(7,16)(8,21)(9,25)(10,15)(11,18)(12,23)(13,20)(14,26)(17,27)(19,28)(22,35)(24,36)(29,33)(30,32)(31,34);(5,7,9,18,10,20,26,19)(6,11,14,16,15,28,25,13)(8,22,23,34,24,17,32,33)(12,27,21,31,30,35,36,29);(5,8,9,23,10,24,26,32)(6,12,14,21,15,30,25,36)(7,17,19,34,20,22,18,33)(11,27,13,29,28,35,16,31);(2,4)(5,7)(6,13)(8,17)(9,19)(10,20)(11,25)(12,29)(14,28)(15,16)(18,26)(21,35)(22,24)(23,34)(27,36)(30,31)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 330029);