#7689890

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

7345 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)(8,12)(9,11)(13,14,18,21)(15,19,23,16)(17,20,26,28)(22,25,27,24);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,14,18,21)(15,19,23,16)(17,20,26,28)(22,25,27,24);(6,11)(10,12)(13,15,18,23)(14,16,21,19)(17,22,26,27)(20,24,28,25);(5,6)(7,10)(8,12)(9,11)(13,15)(14,19)(16,21)(17,22)(18,23)(20,25)(24,28)(26,27);(5,8,9,7)(6,12,11,10)(13,16,18,19)(14,15,21,23)(17,25,26,24)(20,27,28,22);(13,17)(14,20)(15,22)(16,24)(18,26)(19,25)(21,28)(23,27)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=i,e^2=h,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, 7597086);