#7625892

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×#3481	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7848 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)(13,14,19,21)(15,24,26,20)(16,22,25,18)(17,23,27,28);(5,6)(7,10)(8,11)(9,12)(13,15,19,26)(14,20,21,24)(16,17,25,27)(18,23,22,28);(5,7,9,8)(6,10,12,11)(14,21)(18,22)(20,24)(23,28);(13,16)(14,22)(15,17)(18,21)(19,25)(20,28)(23,24)(26,27);(13,17)(14,23)(15,25)(16,26)(18,24)(19,27)(20,22)(21,28);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,18,19,22)(14,16,21,25)(15,23,26,28)(17,24,27,20)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gh,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, 7533088);