#7627265

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×#3475	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6800 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:

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

Polycyclic presentation:

2^9|a^2=i,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534461);