#7683375

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

7074 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)(7,11)(10,12)(13,14)(15,20)(16,22)(17,18)(19,21)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15)(14,20)(16,23)(17,26)(18,25)(19,24)(21,28)(22,27);(5,7)(6,10)(8,12)(9,11)(14,21)(15,24)(16,26)(18,27);(5,8,9,6)(7,10,11,12)(13,16,19,26)(14,22,21,25)(15,23,24,17)(18,20,27,28);(13,17,19,23)(14,18,21,27)(15,16,24,26)(20,22,28,25);(13,18)(14,23)(15,25)(16,20)(17,21)(19,27)(22,24)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7590571);