#7849802

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

7351 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)(13,14)(15,20)(16,18)(17,22)(19,21)(23,24)(25,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,22)(14,20,21,17)(16,23,25,27)(18,24,28,26);(5,7)(6,9)(8,12)(10,11)(14,21)(15,22)(16,25)(24,26);(13,16)(14,18)(15,23)(17,26)(19,25)(20,24)(21,28)(22,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,26)(18,23)(19,20)(21,22)(24,25)(27,28);(5,6)(7,9)(8,11)(10,12)(13,18)(14,16)(15,24)(17,27)(19,28)(20,23)(21,25)(22,26)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7756998);