#7625896

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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7190 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)(6,9)(10,12)(13,14,19,18)(15,20,25,21)(16,22,24,26)(17,23,28,27);(5,6,8,9)(7,10,11,12)(13,15,19,25)(14,20,18,21)(16,17,24,28)(22,23,26,27);(13,15,19,25)(14,21,18,20)(16,17,24,28)(22,27,26,23);(13,16)(14,22)(15,17)(18,26)(19,24)(20,23)(21,27)(25,28);(13,17)(14,23)(15,24)(16,25)(18,27)(19,28)(20,26)(21,22);(1,3,4,2)(5,7,8,11)(6,10,9,12)(13,18,19,14)(15,21,25,20)(16,26,24,22)(17,27,28,23)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,f^2=gi,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, 7533092);