#428076

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

Polycyclic presentation:

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

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, 335272);