#7966827

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

8821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^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, #72304580, #72304587
maximal normal subgroup of	#72304527, #72304580, #72304587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874023);