#7625751

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
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×#3516	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7078 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)(6,9)(10,12)(13,14)(15,21)(16,20)(17,19)(18,22)(23,27)(24,26)(25,28);(5,6,8,9)(7,10,11,12)(13,14,18,22)(15,19,17,21)(16,20,25,28)(23,26,24,27);(13,15,18,17)(14,19,22,21)(16,24,25,23)(20,27,28,26);(13,16,18,25)(14,20,22,28)(15,23,17,24)(19,26,21,27);(1,3,4,2)(13,14)(15,19)(16,20)(17,21)(18,22)(23,26)(24,27)(25,28);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,21,22,19)(16,24,25,23)(20,27,28,26)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532947);