#7626676

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×#3505	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7456 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)(7,8)(9,10)(11,12)(13,14)(15,20)(16,26)(17,27)(18,21)(19,25)(22,28)(23,24);(1,3,4,2)(6,10)(7,11)(13,15)(14,19)(16,23)(17,24)(18,25)(20,21)(22,27)(26,28);(5,7,9,11)(6,8,10,12)(13,16)(14,17)(15,22)(18,27)(19,28)(20,24)(21,26)(23,25);(5,8)(6,11)(7,10)(9,12)(13,16)(14,17)(15,23)(18,27)(19,24)(20,28)(21,26)(22,25);(13,15)(14,20)(16,22)(17,24)(18,25)(19,21)(23,27)(26,28);(13,17)(14,16)(15,24)(18,26)(19,23)(20,22)(21,27)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,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, 7533872);