#7654751

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

7544 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)(14,19)(15,21)(22,27)(23,28);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11);(5,7,9,10)(6,8,11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,25)(24,28)(26,27);(13,15,18,21)(14,20,19,16)(17,23,26,28)(22,25,27,24);(5,6)(7,8)(9,11)(10,12)(13,16,18,20)(14,21,19,15)(17,25,26,24)(22,23,27,28);(5,8)(6,10)(7,11)(9,12)(13,17)(14,22)(15,23)(16,24)(18,26)(19,27)(20,25)(21,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561947);