#7627613

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×#3498	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=h,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, 7534809);