#7625634

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

7428 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)(7,12)(8,10)(9,11)(13,14)(15,21)(16,24)(17,20)(18,23)(19,25)(22,26)(27,28);(5,7,9,8)(6,10,11,12)(13,15,17,23)(14,18,20,21)(16,22,25,28)(19,26,24,27);(13,16)(14,19)(15,22)(17,25)(18,26)(20,24)(21,27)(23,28);(16,25)(19,24)(22,28)(26,27);(1,3,4,2)(14,20)(18,21)(19,24)(26,27);(5,8,9,7)(6,10,11,12)

Polycyclic presentation:

2^9|a^2=i,b^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, 7532830);