#7682663

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×#3494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7155 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)(5,6,8,10)(7,9,12,11)(13,14,17,19)(15,18,21,26)(16,20,24,27)(22,25,23,28);(1,2,3,4)(13,15)(14,18)(16,23)(17,21)(19,26)(20,28)(22,24)(25,27);(5,6)(7,11)(8,10)(9,12)(14,19)(15,21)(20,27)(22,23);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,21)(19,26)(20,28)(22,24)(25,27);(13,16,17,24)(14,20,19,27)(15,22,21,23)(18,25,26,28);(13,14)(15,18)(16,20)(17,19)(21,26)(22,25)(23,28)(24,27)

Polycyclic presentation:

2^9|a^2=h,b^2=g,e^2=i,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, 7589859);