#428072

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1227 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:

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

Polycyclic presentation:

2^9|b^2=ei,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 335268);