#7720377

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×#3546	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7044 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)(5,6,9,7)(8,11,12,10)(13,14,18,21)(15,16,19,23)(17,25,26,20)(22,28,27,24);(1,2,3,4)(15,19)(16,23)(22,27)(24,28);(6,7)(10,11)(13,15)(14,16)(17,22)(18,19)(20,24)(21,23)(25,28)(26,27);(5,7,9,6)(8,11,12,10);(5,8)(6,10)(7,11)(9,12)(13,16)(14,19)(15,21)(17,24)(18,23)(20,27)(22,25)(26,28);(13,17,18,26)(14,20,21,25)(15,22,19,27)(16,24,23,28)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7627573);