#7684312

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
prime radical	#34655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7273 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)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,27)(26,28);(1,2)(3,4)(13,15,18,22)(14,19,20,17)(16,23,25,28)(21,24,27,26);(6,9)(10,12)(14,20)(15,22)(21,27)(23,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,26)(18,25)(19,24)(20,27)(22,28);(13,17,18,19)(14,22,20,15)(16,24,25,26)(21,23,27,28)

Polycyclic presentation:

2^9|b^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7591508);