#7625813

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×#3521	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7416 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)(15,21)(16,22)(17,23)(18,20)(19,25)(24,27)(26,28);(5,6,8,9)(7,10,11,12)(13,15)(14,19)(16,24)(17,22)(18,25)(20,21)(23,26)(27,28);(1,3,4,2)(14,20)(17,27)(19,21)(22,28);(13,15)(14,21)(16,23)(17,22)(18,25)(19,20)(24,26)(27,28);(13,16)(14,22)(15,24)(17,19)(18,26)(20,28)(21,27)(23,25);(5,7,8,11)(6,10,9,12)(13,17,18,27)(14,23,20,24)(15,22,25,28)(16,19,26,21)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,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, 7533009);