#191331

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2283
prime radical	C₂⁴	#117
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,7,5,8,3,6)(9,10)(11,20,23,16)(12,17)(13,18)(14,19)(15,22,29,37)(21,35,24,30)(25,32)(26,33)(27,34)(28,36,31,38)(39,40);(9,11)(10,15,19,29)(12,24)(13,22)(14,23)(16,18,20,34)(17,31,33,28)(21,26)(25,38)(27,37)(30,40,35,32)(36,39);(1,3,5,4)(2,6,8,7)(9,11)(10,16)(12,21)(13,22)(14,23)(15,18)(17,30)(19,20)(24,26)(25,36)(27,37)(28,32)(29,34)(31,40)(33,35)(38,39);(9,12,13,25,14,26,27,39)(10,17,18,32,19,33,34,40)(11,21,22,36,23,24,37,38)(15,28,20,35,29,31,16,30)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dh,g^b=gh,g^c=gh

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239584, #72239591, #72307233
maximal normal subgroup of	#72239584, #72239591, #72307233

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98527);