#7627304

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3522	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7162 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=hi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534500);