#7679706

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	#35518	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6815 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,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, 7586902);