#7625745

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×#3514	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7650 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 7532941);