#246992

order	512
nilpotency class	3
isoclinism	isoclinic class of #246987
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1387 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fhi,g^a=gh,c^b=cg,e^b=eh,f^b=fhi,e^c=ei,f^c=fhi,e^d=eh,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310908, #72340417, #72340421
maximal normal subgroup of	#72310908, #72340417, #72340421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154188);