#423528

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1570 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:

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

Polycyclic presentation:

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

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, 330724);