#7985574

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

8592 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)(5,6,9,8)(7,11,12,10)(13,14,18,20)(15,23,24,19)(16,27,26,21)(17,22,25,28);(1,2)(3,4)(6,8)(10,11)(13,15,18,24)(14,19,20,23)(16,25,26,17)(21,28,27,22);(5,6,9,8)(7,10,12,11)(14,20)(15,24)(16,26)(22,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,28)(26,27);(5,7)(6,11)(8,10)(9,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,28)(20,27)(22,23);(5,8,9,6)(7,11,12,10)(13,17,18,25)(14,22,20,28)(15,26,24,16)(19,27,23,21)

Polycyclic presentation:

2^9|a^2=i,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	#72304526, #72304532, #72304555, …
maximal normal subgroup of	#72304526, #72304532, #72304555, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892770);