#7947781

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7654 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72297211
maximal normal subgroup of	#72297211
maximal quotient of	#72297241, #72297488, #72297950

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7854977);