#7681055

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

7171 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7588251);