#7685733

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	#34681	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7049 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)(5,6,8,10)(7,9,11,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,25)(24,28)(26,27);(5,6)(7,9)(8,10)(11,12)(13,15,18,21)(14,19,20,16)(17,23,26,28)(22,25,27,24);(2,4)(14,20)(15,21)(22,27)(23,28);(1,2)(3,4)(13,14)(15,19)(16,21)(17,22)(18,20)(23,25)(24,28)(26,27);(5,7,8,11)(6,9,10,12)(13,16,18,19)(14,21,20,15)(17,25,26,24)(22,23,27,28);(13,17,18,26)(14,22,20,27)(15,23,21,28)(16,24,19,25)

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=gi,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, 7592929);