#7626723

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×#3505	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6932 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)(7,15)(8,12)(9,10)(11,14)(13,20)(16,19)(17,18)(21,22,27,31)(23,26,25,28)(24,30,35,29)(32,36,33,34);(1,3,4,2)(21,23,27,25)(22,28,31,26)(24,33,35,32)(29,34,30,36);(5,7,11,8)(6,12,14,15)(9,16,13,18)(10,17,20,19)(21,24)(22,29)(23,32)(25,33)(26,34)(27,35)(28,36)(30,31);(5,8,11,7)(6,12,14,15)(9,16,13,18)(10,19,20,17)(21,25,27,23)(22,26,31,28)(24,33,35,32)(29,34,30,36);(5,9,11,13)(6,10,14,20)(7,16,8,18)(12,17,15,19)(21,24)(22,30)(23,33)(25,32)(26,34)(27,35)(28,36)(29,31);(5,10)(6,13)(7,17)(8,19)(9,14)(11,20)(12,18)(15,16)(21,26)(22,23)(24,34)(25,31)(27,28)(29,32)(30,33)(35,36)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=h,e^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, 7533919);