#7697530

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	C₂×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6909 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)(7,11,12,9)(13,14,18,21)(15,22,25,19)(16,17,20,26)(23,28,27,24);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,27)(21,22)(26,28);(2,4)(13,15,18,25)(14,19,21,22)(16,23,20,27)(17,24,26,28);(1,2)(3,4)(13,15)(14,19)(16,23)(17,24)(18,25)(20,27)(21,22)(26,28);(13,16)(14,17)(15,23)(18,20)(19,24)(21,26)(22,28)(25,27);(13,17)(14,20)(15,24)(16,21)(18,26)(19,27)(22,23)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7604726);