#7691554

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

7337 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)(5,6,9,12)(7,10,8,11)(13,14)(15,25)(16,22)(17,23)(18,24)(19,21)(20,26)(27,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,27)(17,24)(18,28)(19,26)(21,25)(22,23);(2,4)(14,21)(15,26)(16,24)(23,28);(1,2)(3,4)(13,16,19,24)(14,22,21,18)(15,27,26,17)(20,23,25,28);(5,7,9,8)(6,10,12,11)(13,17,19,27)(14,23,21,28)(15,24,26,16)(18,20,22,25);(5,8,9,7)(6,11,12,10)(13,18,19,22)(14,24,21,16)(15,28,26,23)(17,25,27,20)

Polycyclic presentation:

2^9|a^2=gi,d^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, 7598750);