#7690874

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

6821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=h,e^2=hi,f^2=i,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, 7598070);