#7984024

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

8511 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)(7,11)(8,10)(9,12)(15,23)(20,26)(24,28)(25,27);(1,2)(3,4)(13,14,18,19)(15,20,23,26)(16,21,22,17)(24,27,28,25);(5,6,8,10)(7,9,12,11)(14,19)(16,22)(20,26)(24,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,21)(15,24)(17,19)(18,22)(20,27)(23,28)(25,26);(13,17,18,21)(14,22,19,16)(15,25,23,27)(20,28,26,24)

Polycyclic presentation:

2^9|b^2=i,c^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	#72304526, #72304536, #72304549
maximal normal subgroup of	#72304526, #72304536, #72304549

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7891220);