#185284

order	512
nilpotency class	3
isoclinism	isoclinic class of #185279
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62826	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2240 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(2,7,10,21)(3,12)(4,16)(8,22,25,31)(9,19,11,17)(14,29)(15,27)(18,28)(20,24,32,23)(26,30);(1,2)(3,8)(4,17)(5,9)(6,11)(7,18)(10,13)(12,23)(14,22)(15,24)(16,21)(19,28)(20,26)(25,27)(29,31)(30,32);(3,12)(4,16)(7,19)(8,23)(14,29)(15,27)(17,21)(18,28)(20,32)(22,31)(24,25)(26,30);(1,3,13,27)(2,8,10,25)(4,14,28,29)(5,15,6,12)(7,20,21,32)(9,24,11,23)(16,30,18,26)(17,22,19,31)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72272564, #72272573
maximal normal subgroup of	#72272564, #72272573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 92480);