#7647710

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

7066 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)(7,9)(8,11)(10,12)(16,24)(20,27)(22,28)(25,26);(1,2)(3,4)(6,7)(10,11)(13,14)(15,19)(16,20)(17,21)(18,23)(22,26)(24,27)(25,28);(5,7,9,6)(8,11,12,10)(13,15)(14,18)(16,22)(17,23)(19,21)(20,25)(24,28)(26,27);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(13,14)(15,18)(16,20)(17,21)(19,23)(22,25)(24,27)(26,28);(13,16,17,24)(14,20,21,27)(15,22,23,28)(18,25,19,26)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=h,f^2=hi,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, 7554906);