#7657441

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

7108 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)(5,6,9,12)(7,8,11,10)(13,14,18,17)(15,21,22,19)(16,24,25,20)(23,26,28,27);(1,2,3,4)(15,22)(19,21)(23,28)(26,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,23)(17,21)(18,22)(20,26)(24,27)(25,28);(13,16,18,25)(14,20,17,24)(15,23,22,28)(19,26,21,27);(5,7)(6,8)(9,11)(10,12)(13,14,18,17)(15,19,22,21)(16,20,25,24)(23,26,28,27);(5,8)(6,11)(7,12)(9,10)(13,17,18,14)(15,21,22,19)(16,24,25,20)(23,27,28,26)

Polycyclic presentation:

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

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, 7564637);