#7985299

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

8222 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)(7,8)(9,12)(10,11)(13,14,19,21)(15,25,26,20)(16,18,24,22)(17,27,28,23);(1,2)(3,4)(6,11)(7,9)(13,15)(14,20)(16,23)(17,22)(18,28)(19,26)(21,25)(24,27);(5,7,10,9)(6,8,11,12)(14,21)(15,26)(16,24)(17,28);(5,8)(6,9)(7,11)(10,12)(13,16,19,24)(14,22,21,18)(15,23,26,27)(17,25,28,20);(5,8)(6,7)(9,11)(10,12)(13,17,19,28)(14,23,21,27)(15,22,26,18)(16,25,24,20);(5,9,10,7)(6,12,11,8)(13,18,19,22)(14,24,21,16)(15,17,26,28)(20,23,25,27)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=h,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	#72284677, #72284682, #72284723, …
maximal normal subgroup of	#72284677, #72284682, #72284723, …
maximal quotient of	#72284904

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7892495);