#7967353

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

7237 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)(7,9)(8,11)(10,12)(13,14,19,21)(15,20,16,18)(17,25,26,22)(23,28,24,27);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,19,16)(14,20,21,18)(17,23,26,24)(22,27,25,28);(5,6)(7,10)(8,11)(9,12)(14,21)(15,16)(17,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,16,19,15)(14,20,21,18)(17,24,26,23)(22,27,25,28);(5,7)(6,9)(8,12)(10,11)(13,17)(14,22)(15,23)(16,24)(18,28)(19,26)(20,27)(21,25);(5,6)(7,10)(8,11)(9,12)(13,18,19,20)(14,15,21,16)(17,27,26,28)(22,24,25,23)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7874549);