#7918393

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

7743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7825589);