#7672762

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
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7146 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579958);