#391632

order	512
nilpotency class	3
isoclinism	isoclinic class of #32870
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 298828);