#7625847

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
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×#3494	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7660 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)(5,6,8,10)(7,9,11,12)(13,14,19,21)(15,23,16,20)(17,22,28,18)(24,25,26,27);(5,6)(7,9)(8,10)(11,12)(13,15,19,16)(14,20,21,23)(17,24,28,26)(18,25,22,27);(1,3,4,2)(14,21)(18,22)(20,23)(25,27);(5,7,8,11)(6,9,10,12)(13,16,19,15)(14,20,21,23)(17,24,28,26)(18,27,22,25);(13,17)(14,22)(15,24)(16,26)(18,21)(19,28)(20,27)(23,25);(13,18,19,22)(14,17,21,28)(15,25,16,27)(20,24,23,26)

Polycyclic presentation:

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

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533043);