#7811054

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

7662 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7718250);