#7625872

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×#3475	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7570 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)(5,6,9,12)(7,10,8,11)(13,14,19,18)(15,20,25,24)(16,21,23,26)(17,22,28,27);(5,6)(7,10)(8,11)(9,12)(13,15,19,25)(14,20,18,24)(16,17,23,28)(21,22,26,27);(5,7,9,8)(6,10,12,11)(14,18)(20,24)(21,26)(22,27);(13,16)(14,21)(15,17)(18,26)(19,23)(20,22)(24,27)(25,28);(13,17)(14,22)(15,23)(16,25)(18,27)(19,28)(20,26)(21,24);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,18,19,14)(15,24,25,20)(16,26,23,21)(17,27,28,22)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,f^2=i,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, 7533068);