#7626717

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
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×#3485	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8278 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,23)(16,20)(17,18)(19,22)(21,28)(24,27)(25,26);(1,3,4,2)(6,9)(10,11);(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,22,23)(17,24,21,26)(18,25,28,27);(5,7)(6,10)(8,12)(9,11)(13,16,19,15)(14,20,22,23)(17,24,21,26)(18,27,28,25);(13,17,19,21)(14,18,22,28)(15,24,16,26)(20,25,23,27);(13,18)(14,21)(15,25)(16,27)(17,22)(19,28)(20,26)(23,24)

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533913);