#7627285

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×#3499	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8120 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=g,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, 7534481);