#7671581

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	#35165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7134 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)(17,25)(20,26)(23,28);(5,6,8,9)(7,10,11,12)(13,14,18,19)(15,20,24,26)(16,17,21,25)(22,23,27,28);(2,3)(14,19)(16,21)(20,26)(22,27);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,15,18,24)(14,20,19,26)(16,22,21,27)(17,23,25,28);(13,16,18,21)(14,17,19,25)(15,22,24,27)(20,23,26,28);(13,17)(14,21)(15,23)(16,19)(18,25)(20,27)(22,26)(24,28)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=gh,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7578777);