#7625636

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator	C₂²	C₂⁵×C₄
Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#3482	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7406 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)(5,6)(7,11)(8,12)(9,10)(13,14)(15,21)(16,20)(17,18)(19,23)(22,28)(24,27)(25,26);(6,10)(11,12)(14,18)(19,21)(20,26)(27,28);(5,7,9,8)(6,11,10,12)(13,15,17,23)(14,19,18,21)(16,24,25,22)(20,28,26,27);(5,7,9,8)(6,11,10,12)(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28);(1,3,4,2)(14,18)(19,21)(20,26)(27,28);(5,8,9,7)(6,12,10,11)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532832);