#7627027

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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×#540	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534223);