#7627192

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×#3568	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6596 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gi,c^2=g,d^2=gh,e^2=hi,f^2=gh,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, 7534388);