#7966947

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

8129 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)(5,6,9,10)(7,11,12,8)(13,14,20,25)(15,23,18,21)(16,17,26,22)(19,24,27,28);(1,2)(3,4)(6,10)(7,12)(13,15,20,18)(14,21,25,23)(16,27,26,19)(17,24,22,28);(5,7)(6,11)(8,10)(9,12)(13,16,20,26)(14,22,25,17)(15,19,18,27)(21,24,23,28);(5,8)(6,12)(7,10)(9,11)(13,17,20,22)(14,16,25,26)(15,24,18,28)(19,21,27,23);(5,8)(6,7)(9,11)(10,12)(13,18,20,15)(14,23,25,21)(16,19,26,27)(17,28,22,24);(5,7)(6,11)(8,10)(9,12)(13,19)(14,24)(15,26)(16,18)(17,21)(20,27)(22,23)(25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^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, #72304537, #72304566
maximal normal subgroup of	#72304527, #72304537, #72304566

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874143);