#7920146

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

6823 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)(7,12)(8,10)(9,11)(13,14,18,20)(15,17,22,19)(16,24,26,21)(23,25,28,27);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,22)(14,19,20,17)(16,23,26,28)(21,25,24,27);(6,11)(10,12)(14,20)(15,22)(21,24)(23,28);(5,6)(7,10)(8,12)(9,11)(13,14)(15,19)(16,21)(17,22)(18,20)(23,25)(24,26)(27,28);(5,8,9,7)(6,12,11,10)(13,16)(14,21)(15,23)(17,27)(18,26)(19,25)(20,24)(22,28);(13,17,18,19)(14,22,20,15)(16,25,26,27)(21,23,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7827342);