#7625812

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×#3515	C₂
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)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,24)(14,19,20,26)(16,17,23,25)(21,22,27,28);(1,3,4,2)(14,20)(19,26)(21,27)(22,28);(13,16)(14,21)(15,17)(18,23)(19,22)(20,27)(24,25)(26,28);(13,17)(14,22)(15,23)(16,24)(18,25)(19,27)(20,28)(21,26);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,27)(24,26)(25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,f^2=g,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, 7533008);