#7985107

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

8342 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)(5,6)(7,8)(9,12)(10,11)(13,14)(15,24)(16,18)(17,28)(19,21)(20,25)(22,27)(23,26);(1,2)(3,4)(6,11)(7,9)(13,15,19,25)(14,20,21,24)(16,23,27,28)(17,22,26,18);(5,7,10,9)(6,8,11,12)(14,21)(15,25)(16,27)(17,26);(5,8)(6,9)(7,11)(10,12)(13,16)(14,22)(15,23)(17,24)(18,21)(19,27)(20,26)(25,28);(5,8)(6,7)(9,11)(10,12)(13,17)(14,23)(15,22)(16,24)(18,25)(19,26)(20,27)(21,28);(5,9,10,7)(6,12,11,8)(13,18,19,22)(14,16,21,27)(15,26,25,17)(20,28,24,23)

Polycyclic presentation:

2^9|b^2=h,c^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	#72284541, #72284682, #72284820, …
maximal normal subgroup of	#72284541, #72284682, #72284820, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892303);