#7868231

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

7224 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)(5,6,9,7)(8,11,12,10)(13,14,18,19)(15,16,21,24)(17,20,26,27)(22,23,28,25);(1,2,3,4)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(6,7)(10,11)(14,19)(15,21)(20,27)(22,28);(5,7,9,6)(8,11,12,10)(13,15)(14,16)(17,22)(18,21)(19,24)(20,23)(25,27)(26,28);(5,8)(6,10)(7,11)(9,12)(13,16)(14,15)(17,25)(18,24)(19,21)(20,28)(22,27)(23,26);(13,17)(14,20)(15,22)(16,23)(18,26)(19,27)(21,28)(24,25)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=hi,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, 7775427);