#7627049

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×#3552	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6624 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)(7,12)(8,10)(9,16)(11,15)(13,17)(14,18)(19,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(1,2,4,3)(5,7)(6,11)(8,12)(9,14)(10,15)(13,18)(16,19)(17,20);(5,6,10,8)(7,12,15,11)(9,13,17,16)(14,19,20,18)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36);(5,8,10,6)(7,12,15,11)(9,16,17,13)(14,19,20,18)(21,24,26,22)(23,29,31,27)(25,28,33,32)(30,34,36,35);(1,3,4,2)(5,7)(6,12)(8,11)(9,14)(10,15)(13,19)(16,18)(17,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35);(5,9)(6,13)(7,14)(8,16)(10,17)(11,18)(12,19)(15,20)(21,23)(22,27)(24,29)(25,30)(26,31)(28,34)(32,35)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=h,e^2=i,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, 7534245);