#8005648

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

7323 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7912844);