#7685432

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

8024 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)(7,9,12,11)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(2,4)(5,6)(7,11)(8,10)(9,12)(14,18)(19,25)(20,26)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,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, 7592628);