#7989705

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

7239 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7896901);