#7625711

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	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7118 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)(15,17)(16,20)(18,19)(21,25)(22,24)(23,26)(27,28);(5,6,8,9)(7,10,11,12)(16,23)(20,26)(22,27)(24,28);(1,3,4,2)(5,7,8,11)(6,10,9,12)(14,19)(15,21)(20,26)(22,27);(13,15,18,21)(14,17,19,25)(16,22,23,27)(20,24,26,28);(13,16)(14,20)(15,22)(17,24)(18,23)(19,26)(21,27)(25,28);(13,17)(14,21)(15,19)(16,24)(18,25)(20,27)(22,26)(23,28)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,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, 7532907);