#7985709

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

8326 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)(15,21)(16,18)(17,20)(19,22)(23,28)(24,26)(25,27);(1,2)(3,4)(6,8)(10,11)(13,15)(14,19)(16,21)(17,25)(18,22)(20,28)(23,26)(24,27);(5,6,9,8)(7,10,12,11)(14,16)(15,22)(20,24)(23,25);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,23)(18,22)(20,27)(24,28)(25,26);(5,7)(6,11)(8,10)(9,12)(13,16,18,14)(15,21,22,19)(17,20,26,24)(23,27,25,28);(5,8,9,6)(7,11,12,10)(13,17,18,26)(14,20,16,24)(15,23,22,25)(19,27,21,28)

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,e^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	#72238159, #72238172, #72238237, …
maximal normal subgroup of	#72238159, #72238172, #72238237, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892905);