#7627541

order	512
nilpotency class	2
isoclinism	isoclinic class of #63041
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×#121	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6220 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)(7,14)(8,11)(9,12)(10,13)(15,20)(16,18)(17,19)(22,24)(27,29)(28,32)(34,35);(5,6)(7,14)(8,16)(9,12)(10,13)(11,18)(15,17)(19,20)(21,22,26,24)(23,29,31,27)(25,28,33,32)(30,35,36,34);(5,7,10,12)(6,9,13,14)(8,15,18,19)(11,17,16,20)(21,23,26,31)(22,27,24,29)(25,30,33,36)(28,34,32,35);(5,8)(6,11)(7,15)(9,17)(10,18)(12,19)(13,16)(14,20)(21,24,26,22)(23,29,31,27)(25,32,33,28)(30,35,36,34);(1,3,4,2)(5,6)(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,20)(21,25,26,33)(22,28,24,32)(23,30,31,36)(27,34,29,35);(5,9)(6,12)(7,13)(8,17)(10,14)(11,19)(15,16)(18,20)

Polycyclic presentation:

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

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534737);