#7967551

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

8502 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)(8,9)(10,12)(13,14,18,21)(15,19,23,17)(16,24,26,20)(22,25,28,27);(1,2)(3,4)(6,9)(10,11);(5,6)(7,10)(8,9)(11,12)(13,15,18,23)(14,17,21,19)(16,22,26,28)(20,25,24,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,19)(16,22)(17,21)(18,23)(20,27)(24,25)(26,28);(5,7)(6,11)(8,12)(9,10)(13,16,18,26)(14,20,21,24)(15,22,23,28)(17,25,19,27);(5,6)(7,10)(8,9)(11,12)(13,17,18,19)(14,15,21,23)(16,25,26,27)(20,22,24,28)

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=h,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	#72304527, #72304584, #72304597
maximal normal subgroup of	#72304527, #72304584, #72304597

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874747);