#7987497

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

7334 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)(5,6,8,9)(7,10,12,11)(13,14)(15,21)(16,18)(17,27)(19,22)(20,23)(24,28)(25,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,20,17)(14,21,23,27)(16,26,19,24)(18,25,22,28);(6,9)(10,11)(13,16,20,19)(14,18,23,22)(15,24,17,26)(21,28,27,25);(5,6)(7,10)(8,9)(11,12)(13,17,20,15)(14,21,23,27)(16,26,19,24)(18,28,22,25);(5,6)(7,11)(8,9)(10,12)(13,18)(14,16)(15,25)(17,28)(19,23)(20,22)(21,26)(24,27);(5,7,8,12)(6,10,9,11)(13,19,20,16)(14,22,23,18)(15,26,17,24)(21,25,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308891, #72308905, #72308918, …
maximal normal subgroup of	#72308891, #72308905, #72308918, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7894693);