#7852847

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

7285 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)(7,10)(8,12)(13,14,18,21)(15,19,22,26)(16,17,20,25)(23,24,27,28);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(15,22)(19,26)(23,27)(24,28);(5,7)(6,8)(9,10)(11,12)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(5,6)(7,12)(8,10)(9,11);(5,8,9,12)(6,10,11,7)(13,16)(14,17)(15,23)(18,20)(19,24)(21,25)(22,27)(26,28);(5,6)(7,8)(9,11)(10,12)(13,17)(14,20)(15,24)(16,21)(18,25)(19,27)(22,28)(23,26)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,e^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, 7760043);