#248638

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

1140 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,8)(4,9)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,37)(28,35)(29,36)(30,38)(31,39)(32,40)(33,41)(34,43)(42,47)(44,46)(45,48);(2,7)(4,5)(8,14)(9,10)(11,12)(15,16)(17,19)(18,23)(20,27)(21,28)(22,29)(24,37)(25,35)(26,36)(30,42)(31,43)(32,44)(33,45)(34,39)(38,47)(40,46)(41,48);(1,3,6,13)(2,8,10,16)(4,11,5,12)(7,14,9,15)(17,20,22,31)(18,24,26,39)(19,27,29,43)(21,32,33,30)(23,34,36,37)(25,40,41,38)(28,44,45,42)(35,47,48,46);(1,4)(3,11)(5,6)(7,9)(12,13)(14,15)(17,21,22,33)(18,25,26,41)(19,28,29,45)(20,30,31,32)(23,35,36,48)(24,38,39,40)(27,42,43,44)(34,46,37,47)

Polycyclic presentation:

2^9|c^2=hi,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	#72306412, #72306466, #72306488
maximal normal subgroup of	#72306412, #72306466, #72306488

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 155834);