#7712805

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

7232 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)(5,6,9,10)(7,11,12,8)(13,14,18,15)(16,21,23,19)(17,22,25,20)(24,26,28,27);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14,18,15)(16,19,23,21)(17,20,25,22)(24,26,28,27);(6,10)(8,11)(14,15)(16,23)(20,22)(24,28);(5,8)(6,7)(9,11)(10,12)(13,15,18,14)(16,21,23,19)(17,22,25,20)(24,27,28,26);(5,7)(6,8)(9,12)(10,11)(13,16,18,23)(14,19,15,21)(17,24,25,28)(20,26,22,27);(13,17,18,25)(14,20,15,22)(16,24,23,28)(19,26,21,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7620001);