#7811053

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

7658 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)(5,6,9,7)(8,11,12,10)(13,14,19,18)(15,20,16,23)(17,25,26,21)(22,28,24,27);(1,2)(3,4)(13,15)(14,20)(16,19)(17,24)(18,23)(21,28)(22,26)(25,27);(6,7)(10,11)(14,18)(15,16)(21,25)(22,24);(5,7,9,6)(8,11,12,10)(13,16,19,15)(14,20,18,23)(17,24,26,22)(21,27,25,28);(5,8,9,12)(6,10,7,11)(13,17,19,26)(14,21,18,25)(15,22,16,24)(20,27,23,28);(13,18,19,14)(15,20,16,23)(17,25,26,21)(22,27,24,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7718249);