#248635

order	512
nilpotency class	3
isoclinism	isoclinic class of #248599
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#61706	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1289 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257443, #72257460, #72257469
maximal normal subgroup of	#72257443, #72257460, #72257469
maximal quotient of	#72264222

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155831);