#7985315

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

8430 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)(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,25)(17,18)(19,21)(22,28)(23,27)(24,26);(1,2)(3,4)(6,8)(10,11)(13,15,19,16)(14,20,21,25)(17,23,28,26)(18,27,22,24);(5,6,9,8)(7,10,12,11)(14,21)(15,16)(17,28)(24,27);(5,7)(6,10)(8,11)(9,12)(13,16,19,15)(14,20,21,25)(17,26,28,23)(18,27,22,24);(5,7)(6,11)(8,10)(9,12)(13,17,19,28)(14,22,21,18)(15,23,16,26)(20,24,25,27);(5,8,9,6)(7,11,12,10)(13,18,19,22)(14,17,21,28)(15,24,16,27)(20,26,25,23)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=hi,d^2=h,e^2=h,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	#72284323, #72284454, #72284682, …
maximal normal subgroup of	#72284323, #72284454, #72284682, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892511);