#7688350

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

7065 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)(5,6,10,8)(7,11,9,12)(13,14,19,17)(15,20,24,23)(16,25,18,21)(22,28,26,27);(1,2)(3,4)(5,7,10,9)(6,11,8,12)(13,15)(14,20)(16,26)(17,23)(18,22)(19,24)(21,28)(25,27);(5,7,10,9)(6,12,8,11)(13,16,19,18)(14,21,17,25)(15,22,24,26)(20,27,23,28);(5,8,10,6)(7,12,9,11)(13,17,19,14)(15,23,24,20)(16,21,18,25)(22,27,26,28);(5,9,10,7)(6,12,8,11)(13,15)(14,20)(16,22)(17,23)(18,26)(19,24)(21,27)(25,28);(13,18,19,16)(14,21,17,25)(15,22,24,26)(20,28,23,27)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,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, 7595546);