#7677740

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

7490 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)(5,6,9,10)(7,11,12,8)(13,14)(15,17)(16,21)(18,20)(19,22)(23,26)(24,27)(25,28);(1,2)(3,4)(13,15,18,22)(14,19,20,17)(16,24,25,23)(21,26,28,27);(6,10)(7,12)(14,20)(15,22)(16,25)(26,27);(5,7)(6,11)(8,10)(9,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(13,16,18,25)(14,21,20,28)(15,23,22,24)(17,26,19,27);(5,8,9,11)(6,7,10,12)(13,17,18,19)(14,22,20,15)(16,26,25,27)(21,24,28,23)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,e^2=i,f^2=h,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, 7584936);