#7681054

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×#537	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6750 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)(3,4)(5,6,9,12)(7,10,8,11)(14,20)(18,28)(21,24)(22,25);(5,6)(7,10)(8,11)(9,12)(13,14,19,20)(15,24,16,21)(17,25,27,22)(18,23,28,26);(2,4)(5,7,9,8)(6,10,12,11)(13,15,19,16)(14,21,20,24)(17,26,27,23)(18,25,28,22);(1,2)(3,4)(5,8,9,7)(6,11,12,10)(13,16,19,15)(14,21,20,24)(17,26,27,23)(18,22,28,25);(13,17)(14,22)(15,23)(16,26)(18,24)(19,27)(20,25)(21,28);(13,18,19,28)(14,23,20,26)(15,25,16,22)(17,24,27,21)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,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, 7588250);