#7985937

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

7243 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)(5,6,9,8)(7,10,12,11)(13,14,18,16)(15,21,23,19)(17,20,25,24)(22,27,28,26);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,6,9,8)(7,11,12,10)(14,16)(19,21)(20,24)(26,27);(5,7)(6,11)(8,10)(9,12)(13,16,18,14)(15,21,23,19)(17,24,25,20)(22,27,28,26);(5,7)(6,10)(8,11)(9,12)(13,15)(14,19)(16,21)(17,22)(18,23)(20,26)(24,27)(25,28);(5,8,9,6)(7,10,12,11)(13,17,18,25)(14,20,16,24)(15,22,23,28)(19,26,21,27)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=h,f^2=hi,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, 7893133);