#7846765

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

8301 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,20)(15,19,23,28)(16,17,22,21)(24,27,26,25);(1,2)(3,4)(13,15)(14,19)(16,24)(17,27)(18,23)(20,28)(21,25)(22,26);(6,9)(10,11)(14,20)(15,23)(17,21)(24,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,25)(18,23)(20,28)(21,27)(22,24);(5,7)(6,10)(8,12)(9,11)(13,16,18,22)(14,21,20,17)(15,24,23,26)(19,25,28,27);(13,17,18,21)(14,22,20,16)(15,25,23,27)(19,24,28,26)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7753961);