#2139055

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=h,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278942
maximal normal subgroup of	#72278942
maximal quotient of	#72279050, #72299372, #72299666

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2046251);