#7684661

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
prime radical	#36098	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7576 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591857);