#7627554

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#3494	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 36 points:

(1,2,3,4)(5,6,10,11)(7,14,18,12)(8,9,15,13)(16,19,20,17)(21,22,27,26)(23,31,32,28)(24,29,34,35)(25,30,33,36);(6,11)(8,15)(12,14)(16,20)(23,32)(24,34)(28,31)(29,35);(5,7)(6,12)(8,16)(9,17)(10,18)(11,14)(13,19)(15,20)(21,23)(22,28)(24,33)(25,34)(26,31)(27,32)(29,30)(35,36);(5,8,10,15)(6,13,11,9)(7,16,18,20)(12,19,14,17)(21,24,27,34)(22,29,26,35)(23,33,32,25)(28,30,31,36);(5,7)(6,14)(8,16)(9,19)(10,18)(11,12)(13,17)(15,20)(21,25,27,33)(22,30,26,36)(23,34,32,24)(28,29,31,35);(5,9,10,13)(6,15,11,8)(7,17,18,19)(12,20,14,16)(21,26,27,22)(23,31,32,28)(24,35,34,29)(25,30,33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534750);