#7625831

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

11080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,f^2=h,b^a=bg,c^a=ch,f^c=fh,e^d=eh

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284589, #72284641, #72285241
maximal normal subgroup of	#72284589, #72284641, #72285241

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7533027);