#7689299

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

7414 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)(13,14,19,21)(15,24,25,20)(16,22,26,18)(17,23,27,28);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,17)(18,23)(19,25)(21,24)(22,28)(26,27);(6,11)(10,12)(14,21)(15,25)(16,26)(23,28);(5,6)(7,10)(8,12)(9,11)(13,16)(14,22)(15,17)(18,21)(19,26)(20,28)(23,24)(25,27);(5,8,9,7)(6,12,11,10)(13,17)(14,23)(15,16)(18,20)(19,27)(21,28)(22,24)(25,26);(13,18,19,22)(14,16,21,26)(15,23,25,28)(17,24,27,20)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,e^2=hi,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, 7596495);