#7866419

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

7222 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)(15,16)(17,21)(18,19)(20,24)(22,23)(25,28)(26,27);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(13,14,18,19)(15,16,20,24)(17,21,26,27)(22,23,28,25);(5,6,9,12)(7,8,11,10)(14,19)(15,20)(21,27)(22,28);(13,15,18,20)(14,16,19,24)(17,22,26,28)(21,23,27,25);(5,7)(6,8)(9,11)(10,12)(13,16)(14,20)(15,19)(17,25)(18,24)(21,22)(23,26)(27,28);(5,8)(6,11)(7,12)(9,10)(13,17)(14,21)(15,22)(16,23)(18,26)(19,27)(20,28)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7773615);