#7683594

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

7031 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7590790);