#7699074

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	#37653	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6728 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606270);