#7853325

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

7165 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)(6,8)(10,11)(13,14,20,24)(15,21,17,19)(16,23,18,22)(25,27,26,28);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,20,17)(14,21,24,19)(16,26,18,25)(22,27,23,28);(5,6,9,8)(7,11,12,10)(13,16,20,18)(14,22,24,23)(15,25,17,26)(19,27,21,28);(5,7,9,12)(6,10,8,11)(13,17,20,15)(14,19,24,21)(16,25,18,26)(22,28,23,27);(5,8,9,6)(7,11,12,10)(13,18,20,16)(14,23,24,22)(15,25,17,26)(19,27,21,28);(13,19)(14,15)(16,27)(17,24)(18,28)(20,21)(22,25)(23,26)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=h,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, 7760521);