#7647188

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

7104 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)(7,12)(8,10)(9,11)(14,19)(15,23)(21,22)(24,25);(1,2)(3,4)(14,19)(20,26)(21,22)(27,28);(5,7,9,8)(6,10,11,12)(13,14)(15,20)(16,21)(17,22)(18,19)(23,26)(24,27)(25,28);(6,11)(10,12)(13,15,18,23)(14,20,19,26)(16,24,17,25)(21,27,22,28);(5,6)(7,10)(8,12)(9,11)(13,16,18,17)(14,21,19,22)(15,24,23,25)(20,27,26,28);(5,8,9,7)(6,12,11,10)(13,17,18,16)(14,22,19,21)(15,25,23,24)(20,28,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7554384);