#7626864

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×#3493	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6636 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)(5,6,10,9)(7,13,15,11)(8,12,16,17)(14,19,20,18)(21,22,28,26)(23,32,31,27)(24,34,36,29)(25,33,35,30);(5,6,10,9)(7,11,15,13)(8,12,16,17)(14,18,20,19)(21,23)(22,27)(24,30)(25,29)(26,32)(28,31)(33,36)(34,35);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,20)(17,19)(21,24)(22,29)(23,33)(25,32)(26,34)(27,35)(28,36)(30,31);(5,8)(6,12)(7,14)(9,17)(10,16)(11,18)(13,19)(15,20)(21,25,28,35)(22,30,26,33)(23,29,31,34)(24,32,36,27);(1,3,4,2)(5,9,10,6)(7,13,15,11)(8,12,16,17)(14,18,20,19)(21,26,28,22)(23,32,31,27)(24,34,36,29)(25,30,35,33);(5,7)(6,11)(8,14)(9,13)(10,15)(12,18)(16,20)(17,19)(21,27,28,32)(22,31,26,23)(24,35,36,25)(29,30,34,33)

Polycyclic presentation:

2^9|a^2=i,b^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, 7534060);