#7625808

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×#534	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7482 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)(6,9)(10,12)(13,14)(15,22)(16,20)(17,18)(19,21)(23,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15,19,16)(14,20,21,22)(17,23,27,25)(18,24,28,26);(1,3,4,2)(14,21)(18,28)(20,22)(24,26);(5,7,8,11)(6,10,9,12)(13,16,19,15)(14,20,21,22)(17,23,27,25)(18,26,28,24);(13,17)(14,18)(15,23)(16,25)(19,27)(20,24)(21,28)(22,26);(13,18)(14,17)(15,24)(16,26)(19,28)(20,23)(21,27)(22,25)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^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, 7533004);