#7684662

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	#34655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7126 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)(3,4)(6,10)(11,12)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,26)(27,28);(5,6)(7,11)(8,12)(9,10)(13,15,18,25)(14,19,22,23)(16,21,26,28)(17,24,27,20);(2,4)(13,14,18,22)(15,23,25,19)(16,20,26,24)(17,21,27,28);(1,2)(3,4)(13,14)(15,19)(16,20)(17,21)(18,22)(23,25)(24,26)(27,28);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,21)(17,19)(18,26)(22,24)(23,27)(25,28);(5,8,9,7)(6,12,10,11)(13,17)(14,21)(15,24)(16,23)(18,27)(19,26)(20,25)(22,28)

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=g,f^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, 7591858);