#7686817

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

7065 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)(13,14,19,21)(15,24,25,20)(16,18,23,28)(17,22,27,26);(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,20,21,24)(16,17,23,27)(18,26,28,22);(2,4)(14,21)(15,25)(16,23)(22,26);(1,2)(3,4)(13,16,19,23)(14,18,21,28)(15,17,25,27)(20,26,24,22);(13,17)(14,22)(15,23)(16,25)(18,20)(19,27)(21,26)(24,28);(13,18)(14,23)(15,26)(16,21)(17,24)(19,28)(20,27)(22,25)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^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, 7594013);