#7918997

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

7116 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,16,21,24)(17,20,25,26)(22,23,27,28);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12);(5,6,9,8)(7,11,12,10)(14,19)(15,21)(20,26)(22,27);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,23)(18,24)(19,21)(20,22)(25,28)(26,27);(5,7)(6,11)(8,10)(9,12)(13,17,18,25)(14,20,19,26)(15,22,21,27)(16,23,24,28)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,e^2=i,f^2=hi,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7826193);